holds for all \(a,b,c\in B\)  (where \(a^-\)  is the inverse of \(a\)  in \((B,\circ )\) ). In [1]} these structures are simply called left semi-braces: here, we will use the prefix left cancellative to distinguish them from the left semi-braces studied in [2]}, where the left cancellativity of \((B,+)\)  is not required. If \((B,+)\)  is a group, then \((B,+,\circ )\)  is a skew left brace, and if in addition \((B,+)\)  is abelian then \((B,+,\circ )\)  is a left brace. For further generalizations of left braces, we refer the reader to [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [2]} and related references.
